Even though the proposed method is particularly well suited for low Mach number flows, we have also tested its behaviour also in a situation in which shock waves occur. For this purpose, we have first considered the classical Sod shock tube problem for an ideal gas proposed in [1]} and also discussed in [2]}. It consists of a right-moving shock wave, an intermediate contact discontinuity and a left-moving rarefaction fan. In this higher Mach number regime it is more appropriate to use the Local Lax-Friedrichs flux (LLF), defined by setting
\(\lambda ^{(n,1)} &= \max \left(\left\Vert \mathbf {u}^{(n,1)^{+}}\right\Vert  + \frac{1}{Ma}c^{(n,1)^{+}},\left\Vert \mathbf {u}^{(n,1)^{-}}\right\Vert  + \frac{1}{Ma}c^{(n,1)^{-}}\right) \\\lambda ^{(n,2)} &= \max \left(\left\Vert \mathbf {u}^{(n,2)^{+}}\right\Vert  + \frac{1}{Ma}c^{(n,2)^{+}},\left\Vert \mathbf {u}^{(n,2)^{-}}\right\Vert  + \frac{1}{Ma}c^{(n,2)^{-}}\right)\) 
